Question

PROVE THAT (trigno)​

Answer 1

           

$\sf{Identities used:} \\ \\ \\ \sin\alpha+\sin\beta=2\sin(\frac{\alpha+\beta}{2}) \cdot \cos(\frac{\alpha-\beta}{2}) \\ \\ \\ \cos\alpha+\cos\beta=2\cos(\frac{\alpha+\beta}{2}) \cdot \cos(\frac{\alpha-\beta}{2})$

$\ \ \ \ \ \boxed{ LHS } \\ \\ \hookrightarrow\ \boxed{\frac{\cos 4A+\cos 3A+\cos 2A}{\sin 4A+\sin 3A + \sin 2A}} \\ \\ \\ \hookrightarrow\ \boxed{\frac{(\cos 4A+\cos 2A)+\cos 3A}{(\sin 4A+\sin 2A)+\sin 3A}} \\ \\ \\ \hookrightarrow\ \boxed{\frac{(2\cos(\frac{4A+2A}{2}) \cdot \cos(\frac{4A-2A}{2}))+\cos3A}{(2\sin(\frac{4A+2A}{2}) \cdot \cos(\frac{4A-2A}{2}))+\sin3A}}$

$\hookrightarrow\ \boxed{\frac{2 \cdot \cos3A \cdot \cos A+\cos3A}{2 \cdot \sin3A \cdot \cos A +\sin3A}} \\ \\ \\ \hookrightarrow\ \boxed{\frac{\cos3A(2\cos A+1)}{\sin3A(2\cos A+1)}} \\ \\ \\ \hookrightarrow\ \boxed{\frac{\cos3A}{\sin3A}} \\ \\ \\ \hookrightarrow\ \boxed{\cot3A} \\ \\ \hookrightarrow\ \boxed{ RHS }$

$\sf{Hence proved!}$

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